Everyday Calculus: Discovering the Hidden Math All Around Us
“Professor Fernandez is a delightfully quirky writer and his book Everyday Calculus is lighthearted and compelling, connecting mathematics to daily life.”
If you have ever asked yourself, “When am I ever going to use this?” about calculus, the premise of Everyday Calculus is simple. As Oscar Fernandez, professor of mathematics, goes about his day, calculus is shown to play its part. The reader will discover that many seemingly unrelated phenomena are surprisingly related, connected by mathematics.
Professor Fernandez begins his day by waking up to a clock radio. He informs us that during the night his brain’s REM and non-REM sleep activity is a cycle, a trigonometric function. So, too, is the voltage that powers his clock radio. There are other types of functions around him, as well. He points out that the voltage on the power line going into his house decreases as its length increases, and the ratio of voltage to distance forms a rational function. The radio signal that comes in over the air to his radio also incurs a power loss and is another rational function. Inside his radio, electromagnetic waves are converted to sound waves. Sound waves are processed by the brain in decibels, a logarithmic function.
Washing in the shower the professor notices that each water droplet is pulled down by gravity—another rational function, and the force of gravity applies to footballs, Frisbees —any object thrown into the air. Of the different types of functions, the professor wonders, “Is there any order to this chaos? Does my morning consist of chance encounters with different functions, or are they all related somehow?”
At the breakfast table, the professor notices derivatives or an average rate of change with respect to time in the cooling his coffee. He tells us, “Wherever there is change there are derivatives.” Rates of change are also important to the stock market, though they are easier to see when graphed. The derivative is the slope of the curve, and the professor notes there are two equally valid ways to measure functions. The first is by calculation, the second by geometry.
An average rate of change is one thing but how about an instantaneous rate of change? There is a problem in the calculation of an instantaneous rate of change however as direct calculation leads to dividing by zero, something mathematicians try to avoid at all costs. A good mathematician can approach zero however without ever having to reach it. This type of calculation is the limit, and its solution is the derivative. If an instantaneous rate of change were to be determined by geometry instead of by calculation, it would be the line drawn tangent to the function’s slope.
On this particular day it happens to be raining when the professor leaves his house, so he takes his umbrella and tells us that a typical raindrop falls from a height of 13,000 feet and increases in both size and speed as it falls. He wonders, “If both a droplet’s mass and its speed increases as it falls, how come it doesn’t crash right through my umbrella? Why do we survive even one raindrop?”
If the professor were to survive the rain (which he does) only to later catch a cold (which he leaves “up in the air”), calculus would be involved in calculating the spread of infection. Professor Fernandez’ goes ahead and does calculates the probability of catching a cold while stuck in a meeting with 20 coworkers, some of whom are potentially sick. It takes days for a cold to incubate and as Everyday Calculus covers but a single day, the reader has to move on—whether or not the professor has caught a cold may not be known until the sequel.
At lunchtime we find the professor at a restaurant, eating sushi and wondering how long humans can overfish fish our oceans before there are no more fish to catch. He tells us the mathematics of fishing is similar to the mathematics of catching a cold - when most everyone has a cold there will be fewer remaining to catch colds just as when there are fewer fish, remaining fish will be harder to catch. The solution to overfishing may be found in sustainability analysis—limiting the yearly catch will allow fish (and fishing) to survive over the long term. The professor exclaims, “One of the great things about mathematics is that its conclusions are universal.”
Professor Fernandez discovers another mathematical connection. When you purchase a financial stock, your profits are compounded—that is, gains will immediately be reinvested, and compound interest provides exponential growth. The connection is that the mathematics of a successful stock pick is similar to population growth. Exponential growth, whether of stock or of people is unsustainable.
One might get the impression that the professor has a light workload because almost immediately after lunch he takes a break, pointing our attention to the act of filling a cup with hot chocolate. He tells us that if the cup were frustum shaped, that is wider at the bottom than at the top, and also filled by a hot-chocolate dispenser that poured at a fixed rate, there would be an exact formula to tell us how fast the chocolate is rising in the cup—at any time and any height. He does admit that such a formula might not be earth shattering but also notes that the very same mathematics could be used to analyze reservoir levels after rainfall, if perchance one happened to be concerned with flooding.
When our professor is done with his hot chocolate, he crumples his frustum-shaped cup and tosses it into the trashcan. He provides the calculus for his toss too along with another mathematical connection. His cup follows the trajectory of a parabola, equivalent to the mathematics of blood flow in the human circulatory system, as the angles of arterial connections are related to the arteries’ radii and to parabolas. The maximum and minimum of the vertical travel of a trajectory, whether of a tossed cup or of blood flowing through an artery is a problem of optimization and nature’s goal (if nature could be said to have a goal) is one of minimizing energy.
Our professor apparently not only has a light teaching load, he also (no surprise) is a daydreamer. Looking out the window and seeing high power lines, he envisions another minimum energy configuration in the sag of the power lines between towers. Though this sag is also due to the force of gravity, its shape is not a parabola but a catenary. The catenary is the shape that minimizes gravitational stored energy.
The workday (such as it was) now over the professor drives home, and while driving figures out how to optimize his route to save on gas. Given a choice of routes, the problem becomes one of how far to drive on high speed and low speed routes that match his car’s fuel efficiency. Though optimizing his short six-mile commute may not save much, if the professor owned a fleet of delivery trucks, even small savings would add up.
Back home the professor meets up with his wife and heads out to the city for dinner and a movie, this time taking commuter transit rail. On the train he figures out how to calculate the miles traveled by each railcar to provide a maintenance schedule. He does not consider measuring the actual distance traveled by something as simple as an odometer so much as deriving the distance from measuring the rail cars’ speed over time. Though this method is inefficient, it is a classic problem in calculus.
The geometric solution of the speed over time function is the area under the curve, which can be estimated by summing rectangles that fit under the curve. This form of calculation introduces error because straight lines (the tops of rectangles) are imperfect matches for curved functions. Errors can be reduced by using ever-increasing numbers of smaller rectangles and as the number of rectangles increases towards infinity, the sum is no longer an approximation but the exact value.
Summing infinity is not easy to do but clever mathematicians have devised a shortcut, and explanation, the professor introduces the integral and the mean value theorem. We learn that the mean value theorem has a wide range of applications, including catching speeders on limited-access highways without having to actually catch speeders speeding. That is, if you knew the exact time the driver entered and then left the highway, the mean value theorem would show whether or not the driver had ever exceeded the speed limit (though the professor does leave out any consideration of legal issues that might be raised by directly handing out speeding tickets by toll takers from their tollbooths).
The train is delayed because of a problem further down the line, and another idea comes to the professor. If one were on a train that was delayed, what is the probability of being delayed by more than N minutes? In this problem’s solution, actual delays could be used to provide a range of times that would also form a probability density curve. The probability density function is useful not just for calculating train delays but also for calculating wait-times for customer call centers and delivery dates for shipping warehouses.
The delay is only a short one and now in an Indian restaurant the professor considers the calculus inside the thermostat in the Tandori oven cooking his meal. He points out that the oven’s thermostat uses the same mathematics as the thermostat inside an air conditioning system, even though the heater is heating and the cooler is cooling – that any calculation of an average value of a continuous function uses the same mathematics, whether the average is of a temperature, a rainfall, or products sold.
Dinner over, and now at the movies, the professor explains how calculus can be used to pick the best seats. “Best” in this instance means sitting where you don’t strain your neck your neck, and in practice is actually a range of seats. Architects who design movie theaters will try to make as many seats as possible the best seats, and similar kinds of analyses are made for new symphony halls with respect to acoustics.
The day over and the book concluded, this reviewer notes that professor Fernandez has indeed succeeded in doing what he set out to do: show calculus’ relevance to daily life.
Everyday Calculus will not only be found to be understandable by non-mathematicians but will also be found to be quite entertaining. Indeed, not everyone considers the calculus going on inside Tandoori ovens, and they should.